b1, -- Cube test
b2, -- Cube test
b3, -- Cube test
+ barycenter,
c,
+ contains_point,
polynomial,
tetrahedron_properties,
tetrahedron_tests,
import Comparisons ((~=), nearly_ge)
import FunctionValues (FunctionValues(..), empty_values)
import Misc (factorial)
-import Point (Point, scale)
+import Point (Point(..), scale)
import RealFunction (RealFunction, cmult, fexp)
-import ThreeDimensional (ThreeDimensional(..))
data Tetrahedron =
Tetrahedron { function_values :: FunctionValues,
" v3: " ++ (show (v3 t)) ++ "\n"
-instance ThreeDimensional Tetrahedron where
- center (Tetrahedron _ v0' v1' v2' v3' _) =
- (v0' + v1' + v2' + v3') `scale` (1/4)
-
- -- contains_point is only used in tests.
- contains_point t p0 =
- b0_unscaled `nearly_ge` 0 &&
- b1_unscaled `nearly_ge` 0 &&
- b2_unscaled `nearly_ge` 0 &&
- b3_unscaled `nearly_ge` 0
+-- | Find the barycenter of the given tetrahedron.
+-- We just average the four vertices.
+barycenter :: Tetrahedron -> Point
+barycenter (Tetrahedron _ v0' v1' v2' v3' _) =
+ (v0' + v1' + v2' + v3') `scale` (1/4)
+
+-- | A point is internal to a tetrahedron if all of its barycentric
+-- coordinates with respect to that tetrahedron are non-negative.
+contains_point :: Tetrahedron -> Point -> Bool
+contains_point t p0 =
+ b0_unscaled `nearly_ge` 0 &&
+ b1_unscaled `nearly_ge` 0 &&
+ b2_unscaled `nearly_ge` 0 &&
+ b3_unscaled `nearly_ge` 0
+ where
+ -- Drop the useless division and volume calculation that we
+ -- would do if we used the regular b0,..b3 functions.
+ b0_unscaled :: Double
+ b0_unscaled = volume inner_tetrahedron
where
- -- Drop the useless division and volume calculation that we
- -- would do if we used the regular b0,..b3 functions.
- b0_unscaled :: Double
- b0_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v0 = p0 }
+ inner_tetrahedron = t { v0 = p0 }
- b1_unscaled :: Double
- b1_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v1 = p0 }
+ b1_unscaled :: Double
+ b1_unscaled = volume inner_tetrahedron
+ where inner_tetrahedron = t { v1 = p0 }
- b2_unscaled :: Double
- b2_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v2 = p0 }
+ b2_unscaled :: Double
+ b2_unscaled = volume inner_tetrahedron
+ where inner_tetrahedron = t { v2 = p0 }
- b3_unscaled :: Double
- b3_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v3 = p0 }
+ b3_unscaled :: Double
+ b3_unscaled = volume inner_tetrahedron
+ where inner_tetrahedron = t { v3 = p0 }
+{-# INLINE polynomial #-}
polynomial :: Tetrahedron -> (RealFunction Point)
polynomial t =
V.sum $ V.singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `V.snoc`
-- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a
-- capital 'B' in the Sorokina/Zeilfelder paper.
beta :: Tetrahedron -> Int -> Int -> Int -> Int -> (RealFunction Point)
-beta t i j k l
- | (i + j + k + l == 3) =
- coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
- | otherwise = error "basis function index out of bounds"
+beta t i j k l =
+ coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
where
denominator = (factorial i)*(factorial j)*(factorial k)*(factorial l)
coefficient = 6 / (fromIntegral denominator)
-- | The coefficient function. c t i j k l returns the coefficient
-- c_ijkl with respect to the tetrahedron t. The definition uses
-- pattern matching to mimic the definitions given in Sorokina and
--- Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
--- function will simply error.
+-- Zeilfelder, pp. 84-86. If incorrect indices are supplied, the world
+-- will end. This is for performance reasons.
c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
c !t !i !j !k !l =
coefficient i j k l
+ (1/96)*(lt + fl + ft + rt + bt + fr)
+ (1/96)*(fd + ld + bd + br + rd + bl)
- coefficient _ _ _ _ = error "coefficient index out of bounds"
-
-- | Compute the determinant of the 4x4 matrix,
det p0 p1 p2 p3 =
term5 + term6
where
- (x1, y1, z1) = p0
- (x2, y2, z2) = p1
- (x3, y3, z3) = p2
- (x4, y4, z4) = p3
+ Point x1 y1 z1 = p0
+ Point x2 y2 z2 = p1
+ Point x3 y3 z3 = p2
+ Point x4 y4 z4 = p3
term1 = ((x2 - x4)*y1 - (x1 - x4)*y2 + (x1 - x2)*y4)*z3
term2 = ((x2 - x3)*y1 - (x1 - x3)*y2 + (x1 - x2)*y3)*z4
term3 = ((x3 - x4)*y2 - (x2 - x4)*y3 + (x2 - x3)*y4)*z1
-- | Computed using the formula from Lai & Schumaker, Definition 15.4,
-- page 436.
+{-# INLINE volume #-}
volume :: Tetrahedron -> Double
volume t
| v0' == v1' = 0
-- | The barycentric coordinates of a point with respect to v0.
+{-# INLINE b0 #-}
b0 :: Tetrahedron -> (RealFunction Point)
b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
where
-- | The barycentric coordinates of a point with respect to v1.
+{-# INLINE b1 #-}
b1 :: Tetrahedron -> (RealFunction Point)
b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
where
-- | The barycentric coordinates of a point with respect to v2.
+{-# INLINE b2 #-}
b2 :: Tetrahedron -> (RealFunction Point)
b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
where
-- | The barycentric coordinates of a point with respect to v3.
+{-# INLINE b3 #-}
b3 :: Tetrahedron -> (RealFunction Point)
b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
where
[ testCase "volume1" volume1,
testCase "doesn't contain point1" doesnt_contain_point1]
where
- p0 = (0, -0.5, 0)
- p1 = (0, 0.5, 0)
- p2 = (2, 0, 0)
- p3 = (1, 0, 1)
+ p0 = Point 0 (-0.5) 0
+ p1 = Point 0 0.5 0
+ p2 = Point 2 0 0
+ p3 = Point 1 0 1
t = Tetrahedron { v0 = p0,
v1 = p1,
v2 = p2,
doesnt_contain_point1 =
assertEqual "doesn't contain an exterior point" False contained
where
- exterior_point = (5, 2, -9.0212)
+ exterior_point = Point 5 2 (-9.0212)
contained = contains_point t exterior_point
[ testCase "volume1" volume1,
testCase "contains point1" contains_point1]
where
- p0 = (0, -0.5, 0)
- p1 = (2, 0, 0)
- p2 = (0, 0.5, 0)
- p3 = (1, 0, 1)
+ p0 = Point 0 (-0.5) 0
+ p1 = Point 2 0 0
+ p2 = Point 0 0.5 0
+ p3 = Point 1 0 1
t = Tetrahedron { v0 = p0,
v1 = p1,
v2 = p2,
contains_point1 :: Assertion
contains_point1 = assertEqual "contains an inner point" True contained
where
- inner_point = (1, 0, 0.5)
+ inner_point = Point 1 0 0.5
contained = contains_point t inner_point
testCase "doesn't contain point4" doesnt_contain_point4,
testCase "doesn't contain point5" doesnt_contain_point5]
where
- p2 = (0.5, 0.5, 1)
- p3 = (0.5, 0.5, 0.5)
- exterior_point = (0, 0, 0)
+ p2 = Point 0.5 0.5 1
+ p3 = Point 0.5 0.5 0.5
+ exterior_point = Point 0 0 0
doesnt_contain_point2 :: Assertion
doesnt_contain_point2 =
assertEqual "doesn't contain an exterior point" False contained
where
- p0 = (0, 1, 1)
- p1 = (1, 1, 1)
+ p0 = Point 0 1 1
+ p1 = Point 1 1 1
t = Tetrahedron { v0 = p0,
v1 = p1,
v2 = p2,
doesnt_contain_point3 =
assertEqual "doesn't contain an exterior point" False contained
where
- p0 = (1, 1, 1)
- p1 = (1, 0, 1)
+ p0 = Point 1 1 1
+ p1 = Point 1 0 1
t = Tetrahedron { v0 = p0,
v1 = p1,
v2 = p2,
doesnt_contain_point4 =
assertEqual "doesn't contain an exterior point" False contained
where
- p0 = (1, 0, 1)
- p1 = (0, 0, 1)
+ p0 = Point 1 0 1
+ p1 = Point 0 0 1
t = Tetrahedron { v0 = p0,
v1 = p1,
v2 = p2,
doesnt_contain_point5 =
assertEqual "doesn't contain an exterior point" False contained
where
- p0 = (0, 0, 1)
- p1 = (0, 1, 1)
+ p0 = Point 0 0 1
+ p1 = Point 0 1 1
t = Tetrahedron { v0 = p0,
v1 = p1,
v2 = p2,
where
-- | Returns the domain point of t with indices i,j,k,l.
domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
- domain_point t i j k l
- | i + j + k + l == 3 = weighted_sum `scale` (1/3)
- | otherwise = error "domain point index out of bounds"
+ domain_point t i j k l =
+ weighted_sum `scale` (1/3)
where
v0' = (v0 t) `scale` (fromIntegral i)
v1' = (v1 t) `scale` (fromIntegral j)
projects / spline3.git / blobdiff